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Dynamic two-center interference in high-order harmonic generation
from molecules with attosecond nuclear motion
Baker, S.; Robinson, J. S.; Lein, M.; Chirila, C. C.; Torres, R.; Bandulet, H.
C.; Comtois, D.; Kieffer, J. C.; Villeneuve, D. M.; Tisch, J.W. G.; Marangos,
J. P.
Dynamic Two-Center Interference in High-Order Harmonic Generation
from Molecules with Attosecond Nuclear Motion
S. Baker,1J. S. Robinson,1M. Lein,2C. C. Chirila˘,2R. Torres,1H. C. Bandulet,3D. Comtois,3J. C. Kieffer,3 D. M. Villeneuve,4J. W. G. Tisch,1and J. P. Marangos1
1Department of Physics, Imperial College London, London, SW7 2AZ, United Kingdom 2
Institute of Physics, University of Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel, Germany
3INRS-Centre E´ nergie, Mate´riaux et Te´le´communications, 1650 Lionel-Boulet, Varennes, Que´bec, J3X 1S2, Canada 4National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, K1A 0R6, Canada
(Received 24 September 2007; revised manuscript received 22 November 2007; published 30 July 2008) We report a new dynamic two-center interference effect in high-harmonic generation from H2, in which the attosecond nuclear motion of H2 initiated at ionization causes interference to be observed at lower harmonic orders than would be the case for static nuclei. To enable this measurement we utilize a recently developed technique for probing the attosecond nuclear dynamics of small molecules. The experimental results are reproduced by a theoretical analysis based upon the strong-field approximation which incorporates the temporally dependent two-center interference term.
DOI:10.1103/PhysRevLett.101.053901 PACS numbers: 42.65.Ky, 33.15.Dj, 42.65.Sf
High-harmonic generation (HHG) has proven to be a rich area of study over the last decade, finding application in areas such as coherent x-ray production [1,2], attosecond pulse generation [3–5], and time resolved probing of nu-clear dynamics [6,7]. HHG has also led to important ad-vances towards the structural imaging of small molecules [8–15], the harmonic emission depending on the nature of the molecular orbital involved. This is seen most clearly within the strong-field approximation (SFA), in which the amplitude for HHG is determined by the Fourier transform of the bound state wave function.
The wave functions relevant to HHG are those describ-ing the propagated continuum electron ( c), and the bound electronic state from which the electron was ionized ( g). Interference between c and g on recollision of the electron wave packet with its parent ion results in an oscillating dipole (h cjrj gi) being induced on the mole-cule as the probability distribution of the electron density around the nuclei (j c gj2) varies during the
recolli-sion. It is the acceleration of this dipole which is respon-sible for harmonic emission. Within this picture, suppression of harmonic emission occurs if the shape, size, and symmetry of c and g are such that only a weak net oscillating dipole is induced on the molecule. For example, for a diatomic molecule with a symmetric g, the dipole induced at each molecular center oscillates precisely out of phase if 2R cos [8], where R is the molecular internuclear separation, is the angle be-tween the molecular axis and the electric field of the driving laser, and is the de Broglie wavelength of the returning electron. Destructive interference is thus a result of a resonance in the electronic dipole term, being first predicted in H2and H2 molecules in 2002 [9]. To date this
effect has been observed experimentally in CO2 [13,14],
but not in H2.
Here we report the observation of a new kind of two-center interference, in which the nuclear dynamics launched at ionization play a critical role. In previous studies, the molecular nuclei have been considered static, and the chirp of the returning electron wave packet largely ignored. We now show that in a system with fast moving nuclei (H2), the interference occurs in a transient fashion
involving a dynamic matching of the molecular internu-clear separation and the recolliding electron wavelength.
To enable this measurement, we have used a recently developed technique termed PACER (probing attosecond dynamics by chirp encoded recollision) which allows the nuclear dynamics following ionization to be probed with a temporal resolution of roughly 100 as [6]. The essence of PACER is the sensitivity of the harmonic signal to changes in the nuclear part of the wave function R; t during the excursion time of the electron , through the nuclear autocorrelation function c R R; 0R; dR. c (and thus the harmonic signal) decreases the more the nuclei move in the small time interval [6,16]. In addition, since different harmonic orders correspond to different values of [17], a single recollision probes the nuclear wave packet at a range of times. Each harmonic spectrum therefore has within it details of the nuclear motion, which can be revealed on comparison of harmonic signals in different isotopes of a molecular species: it has been shown that the ratio of harmonic emission in D2and H2increases
with harmonic order (or ) [6], since at longer times the
difference in the internuclear separation of H2 and D2
increases.
PACER offers the attosecond resolution necessary to observe dynamic two-center interference. However, in pre-vious measurements the harmonic signal was predomi-nantly sensitive to the nuclear part of the wave function, through c [6]. The electronic contribution was largely insignificant [18] because the measurement was made in a
randomly aligned sample. To allow the observation of dynamic interference, we enhance the sensitivity to the electronic part of the wave function by employing longer and more intense driving pulses, to produce a significant degree of alignment by the driving pulse itself.
This work was conducted at the Advanced Laser Light Source (ALLS) in Quebec, Canada. The 800 nm, 30:0 0:5 fs pulse was diagnosed using a Thales 6800913A au-tocorrelator at a position equivalent to the interaction region. Previous measurements showed the pulse contrast to be 103 within 200 fs of the pulse, and >106 over 10 ps. The beam was focused by an f 400 mm lens beneath a pulsed gas jet, the repetition rate of which was limited to 4 Hz by the pressure in the detection chamber. The gas jet had previously been characterized so as to deliver H2 and
D2 to the interaction region at a constant density (14%).
The confocal parameter of the focused beam was measured as 5.5 mm; the path through the jet is estimated to be 0.5 mm. The focus was positioned 4 mm before the jet to ensure that short electron trajectories dominated the har-monic signal [19] (achieving a unique time to frequency mapping). The rms variation in pulse energy (2%) was monitored during each data run. The generated signal was resolved by a spectrometer, and measured by an imaging MCP, phosphor screen, and CCD camera.
Single-shot spectra were observed for a range of driving field intensities. The on-target intensity was determined from the position of the harmonic cutoff (corresponding to 3:17Up Ip), ensuring that the value stated is that of the part of the beam dominating the emission of the harmonics detected. The error in intensity was found by considering the increase needed to generate one more order than was observed. Intensity values determined in this way are within 30% of those determined from energy and spot size readings. Preliminary measurements were made to ensure that harmonic emission was not saturated.
Figure1shows the measured ratio of harmonic signals in D2 and H2 at two driving field intensities: 3:0 0:1
1014 W cm2 and 2:2 0:2 1014 W cm2, plotted
against the calculated electron travel time corresponding to each harmonic order [16]. The average signal ratio was computed over 500 single-shot spectra in each gas, with error bars representing the standard error. We have con-firmed that comparison of the signal in two experimental runs in the same gas yields a harmonic ratio that is constant with . However, for > 1:5 fs there are large errors, and the ratio of signals in, e.g., H2 : H2 begins to deviate
strongly from a constant value. This is because large are associated with very weak emission in the harmonic cutoff, where the signal to noise ratio is poor, and the ratio is extremely sensitive to the background level used (a 0.2% change in the background level is found to shift the two data points at highest by 10% and 30%, respectively, while leaving other data points unaffected).
Over the range of for which the experimental data is reliable a clear trend is seen: the harmonic ratio increases
with for both driving field intensities, but this increase is more significant at the higher driving field intensity, the difference between the data at the two intensities becoming more pronounced for longer .
The driving field intensity affects HHG through both the momentum evolution of the returning electron wave packet and the alignment distribution of the sample at the peak of the driving pulse. Both of these factors affect the condi-tions for two-center interference, and thus we investigate if this is responsible for the faster growth in harmonic ratio with observed for the higher driving field intensity case. However, in this system we must consider the interference as a dynamic process, occurring when the internuclear separation of the expanding nuclear wave packet passes through a value that matches the returning electron wave-length at that time. We therefore define a simple condition for destructive interference in this dynamic system as 2Rt cosm t, where m is the modal value of the alignment distribution at the peak of the pulse, calcu-lated by the method detailed in [20]. Here sech2 pulses of FWHM 30 fs and peak intensities 3:0 1014 W cm2and
2:2 1014 W cm2 are used to calculate the evolution of under the pulse envelope. The alignment distribution
at the moment of HHG is assumed to be that at the peak of
D
/H harmonic rat
io
2
2
Electron travel time (fs)
D / H harmo n ic r a ti o 2 2 Harmonic order H2 D H D 2 2 2 21 25 29 33 37 41 45 21 25 29 33 37 a) b)
FIG. 1. (a) Measured ratio of harmonic signals in D2and H2at driving field intensity 3:0 0:1 1014W cm2. Two indepen-dent data sets are shown. Black line shows prediction from SFA calculation [Eq. (2)]. Gray line shows SFA calculation in which the nuclear motion has been neglected. Dotted line shows SFA calculation in which two-center interference is neglected. Inset shows calculated alignment distribution [20] at the pulse peak in H2 and D2 at this intensity. For comparison the dotted curve shows for the 8 fs case investigated in Ref. [6]. (b) Same as (a), but for driving field intensity 2:2 0:2 1014W cm2.
the pulse (Fig.1insets). This calculation has been shown to yield accurate results when compared to measured align-ment distributions at similar intensities [21]. We find that the experimental error in intensity leads to an error of only 1% in the value of cos2.
We then plot in Fig.22Rt cosm for H2 (solid line,
where Rt is the average value of the expanding nuclear wave function, calculated as in [16]) and the returning electron wavelength as a function of (data points) for the appropriate experimental conditions, using the relation-ship h2men@!0:5 [13], where m
e is the electron mass, ! is the driving laser frequency, and n is the har-monic order. A point of intersection of these two curves thus represents suppression of H2harmonic emission at the
harmonic order corresponding to the instantaneous elec-tron wavelength at the time of intersection. In Fig. 2we also plot 2Reqcosm, where Reqis the equilibrium
sepa-ration of the H2 molecule (1.4 a.u).
Figure 2 shows that the two driving field intensities employed are indeed in different two-center interference regimes: for the high driving field intensity this simple model predicts that destructive interference may occur in H2 around the 39th harmonic order emitted 1.4 fs after
ionization, whereas for the low intensity case, destructive interference is not predicted to occur. Further, for an in-tensity of 3:0 0:1 1014 W cm2, destructive interfer-ence is not expected to occur if one neglects the nuclear dynamics (dotted line), which are a necessary condition for suppression of H2harmonics to occur. A new dynamic two-center interference effect is therefore introduced, as repre-sented schematically in Fig. 3. If this simple analysis is repeated for D2, one finds that the condition for destructive
interference is not satisfied for our experimental
condi-tions. The signature of the dynamic interference in this system will thus be the observation of higher values for the ratio of harmonic signals in D2 and H2, as emission is
suppressed in H2only. Our measurements (Fig.1) are thus
in qualitative agreement with such an effect, since we detect elevated signal ratios for the high intensity case as compared to the lower intensity case.
We note that destructive two-center interference nor-mally gives a dip in the harmonic spectrum [13] centered around the order corresponding to the wave packet com-ponent which best satisfies 2R cos . For our experi-mental conditions this is the 39th harmonic order (see Fig.2). Since this is very high in the cut-off region of the spectrum, we do not generate enough harmonic orders to clearly observe the dip, and the interference is manifested in our experimental data only as an increase in the ratio of the harmonic signals (D2=H2) for higher orders.
To test further whether a dynamic two-center interfer-ence effect could play a significant role, we perform cal-culations of HHG in a simplified strong-field approxi-mation, including the known internuclear dynamics of H2 and D2, and the effect of two-center interference. If c is described as a superposition of plane waves of amplitudes ak, [ cR akeikxdk], the harmonic signal at frequency ! is proportional to !2jakvkj2[11], where
the recombination amplitude vk in the case of a molecule can be formulated as [22]
vk Z 0Rh0Rjkjeikx0RiR; kdR; (1)
0R and 0R being the electronic ground states of the
neutral molecule and ion, respectively, and 0R and R; the initial and propagated nuclear wave packets
2Rcos( ) m or (A) o
Electron travel time (fs)
FIG. 2. Data points show electron wavelength correspond-ing to the harmonic orders observed at drivcorrespond-ing field inten-sities of 3:0 0:1 1014 W cm2 (black) and 2:2 0:2 1014 W cm2 (gray), plotted against electron travel time. Solid lines show 2Rt cosm for H2
at the high (black) and low (gray) intensities. Dotted line shows 2Reqcosm for H2 at an intensity 3:0 0:1 1014 W cm2. Inset shows SFA calcula-tion of signal ratio at 3:0 1014W cm2 assuming a single alignment angle m.
FIG. 3 (color online). Dynamic two-center interference in HHG, considering a chirped returning electron wave packet (1> 2> 3) and an evolving nuclear wave function. Red and blue represent opposite signs of the wave functions. At early times (A) and late times (C), the condition for two-center interference is not satisfied [2Rt1 cos 1, 2Rt3 cos 3]. The corresponding low and high-order harmonics are there-fore emitted without significant interference. At intermediate times (B), 2Rt2 cos 2, and thus the emission of harmonics of photon energy h=2m2
in the molecular ion. R-independent factors rk can be re-moved as prefactors to this integral to yield vk
rkck with [22]
ck Z 0R R; k cos kR cos=2 dR; (2) which is then averaged over the relevant distribution of alignment angles (calculated as described earlier in text). Assuming isotopic invariance of ak, the ratio of harmonic signals in an isotope pair thus reduces to the modulus squared of the ratio of ck in each isotope. Since the tunneling ionization rates of H2 and D2 (proportional to
exp 22Ip3=2=3E [23]) differ by <3% for these con-ditions, and the shape of ak in atoms has been shown to be insensitive to large changes in Ip [24], isotopic invari-ance of ak is a reasonable assumption. The ratio of harmonic signals calculated by this method is shown in Fig. 1. We also show calculations in which two-center interference is neglected by dropping the cosine term, and in which the nuclei are considered fixed [R;
0R]. To allow for the experimental error in gas density,
all calculations shown in Fig.1have been scaled by a small factor (0.85). In separate calculations we have investigated the effect of the coupling between the laser field and the molecular ion, and found that this has only a small effect on the predicted ratios. Therefore we neglect this effect in the calculations presented in Fig.1.
It can be seen from Fig. 1 that the SFA calculation predicts that higher ratios are obtained for the high inten-sity as compared to the low inteninten-sity case, and that this effect is more pronounced as increases. The SFA calcu-lation is therefore in qualitative agreement with the simple model presented in Fig.2. There is also excellent quanti-tative agreement between the experimental data and the SFA calculation for both intensity regimes. In addition, it can be seen from Fig.1that if either the nuclear motion or the effect of two-center interference are neglected from the calculation, the agreement with the experimental data is lost. This confirms that what we observe is a dynamic two-center interference effect in HHG in H2, in which the
nuclear motion is crucial, resulting in the interference occurring at lower harmonic orders than would be the case if the nuclei were static. For our conditions this is particularly important because it results in the interference occurring at the 39th order, rather than at the 53rd order (which is not generated in this experiment).
We also note that an SFA result considering only the modal value of the alignment distributions (Fig. 2 inset) predicts a peak ratio occurring at a time 1.38 fs after ionization. This agrees well with the time at which the simple model (Fig.2) predicts that destructive interference will occur in H2 (1.4 fs after ionization). However, the
reason for this agreement is not clear, since the SFA result [Eq. (2)] involves an integral over R in which the integrand includes both the evolved and initial nuclear wave packets.
Therefore, while it seems intuitive to use the value of R at the time of recombination in the simple model, it is not clear that this is the correct choice. Whilst we are not at present able to give a theoretical proof for this agreement, we have confirmed by calculation that it is robust with respect to changes in the laser parameters or nuclear mass. Furthermore, we have noted that the agreement is observed only when the ratio of isotopes is taken.
In conclusion, we have observed a new kind of two-center interference by studying HHG in H2 within the
PACER technique. This is a dynamic effect involving an interplay between the nuclear motion and the time-dependent nature of the returning electron wave packet. In essence, the nuclear dynamics cause the interference to occur at lower harmonic orders than would be the case for static nuclei. Thus we have been able to observe recombi-nation interference in H2 for the first time, in a dynamic
manifestation.
This work was supported by EPSRC (Grants No. EP/ C530764/1 and No. EP/E028063/1) and the Deutsche Forschungsgemeinschaft. We also thank the staff at ALLS and the Canadian Foundation for Innovation.
*sarah.baker@imperial.ac.uk
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![Figure 1 shows the measured ratio of harmonic signals in D 2 and H 2 at two driving field intensities: 3:0 0:1 10 14 W cm 2 and 2:2 0:2 10 14 W cm 2 , plotted against the calculated electron travel time corresponding to each harmonic order [16]](https://thumb-eu.123doks.com/thumbv2/123doknet/14164221.473607/3.918.497.816.68.429/figure-measured-harmonic-intensities-calculated-electron-corresponding-harmonic.webp)
